Negatively-refractive focusing and sensing apparatus, methods, and systems

ABSTRACT

Apparatus, methods, and systems provide negatively-refractive focusing and sensing of electromagnetic energy. In some approaches the negatively-refractive focusing includes providing an interior focusing region with an axial magnification substantially greater than one. In some approaches the negatively-refractive focusing includes negatively-refractive focusing with a transformation medium, where the transformation medium may include an artificially-structured material such as a metamaterial.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is related to and claims the benefit of theearliest available effective filing date(s) from the following listedapplication(s) (the “Related Applications”) (e.g., claims earliestavailable priority dates for other than provisional patent applicationsor claims benefits under 35 USC § 119(e) for provisional patentapplications, for any and all parent, grandparent, great-grandparent,etc. applications of the Related Application(s)).

Related Applications

For purposes of the USPTO extra-statutory requirements, the presentapplication constitutes a continuation-in-part of U.S. patentapplication Ser. No. 12/156,443, entitled FOCUSING AND SENSINGAPPARATUS, METHODS, AND SYSTEMS, naming Jeffrey A. Bowers, Roderick A.Hyde, Edward K. Y. Jung, John Brian Pendry, David Schurig, David R.Smith, Clarence T. Tegreene, Thomas A. Weaver, Charles Whitmer, andLowell L. Wood, Jr. as inventors, filed May 30, 2008, which is currentlyco-pending, or is an application of which a currently co-pendingapplication is entitled to the benefit of the filing date.

For purposes of the USPTO extra-statutory requirements, the presentapplication constitutes a continuation-in-part of U.S. patentapplication Ser. No. 12/214,534, entitled EMITTING AND FOCUSINGAPPARATUS, METHODS, AND SYSTEMS, naming Jeffrey A. Bowers, Roderick A.Hyde, Edward K. Y. Jung, John Brian Pendry, David Schurig, David R.Smith, Clarence T. Tegreene, Thomas A. Weaver, Charles Whitmer, andLowell L. Wood, Jr. as inventors, filed Jun. 18, 2008, which iscurrently co-pending, or is an application of which a currentlyco-pending application is entitled to the benefit of the filing date.

For purposes of the USPTO extra-statutory requirements, the presentapplication constitutes a continuation-in-part of U.S. patentapplication Ser. No. (NOT YET ASSIGNED), entitled EMITTING ANDNEGATIVELY-REFRACTIVE FOCUSING APPARATUS, METHODS, AND SYSTEMS, namingJeffrey A. Bowers, Roderick A. Hyde, Edward K. Y. Jung, John BrianPendry, David Schurig, David R. Smith, Clarence T. Tegreene, Thomas A.Weaver, Charles Whitmer, and Lowell L. Wood, Jr. as inventors, filedJul. 25, 2008, which is currently co-pending, or is an application ofwhich a currently co-pending application is entitled to the benefit ofthe filing date.

The United States Patent Office (USPTO) has published a notice to theeffect that the USPTO's computer programs require that patent applicantsreference both a serial number and indicate whether an application is acontinuation or continuation-in-part. Stephen G. Kunin, Benefit ofPrior-Filed Application, USPTO Official Gazette Mar. 18, 2003, availableat http://www.uspto.gov/web/offices/com/sol/og/2003/week11/patbene.htm.The present Applicant Entity (hereinafter “Applicant”) has providedabove a specific reference to the application(s)from which priority isbeing claimed as recited by statute. Applicant understands that thestatute is unambiguous in its specific reference language and does notrequire either a serial number or any characterization, such as“continuation” or “continuation-in-part,” for claiming priority to U.S.patent applications. Notwithstanding the foregoing, Applicantunderstands that the USPTO's computer programs have certain data entryrequirements, and hence Applicant is designating the present applicationas a continuation-in-part of its parent applications as set forth above,but expressly points out that such designations are not to be construedin any way as any type of commentary and/or admission as to whether ornot the present application contains any new matter in addition to thematter of its parent application(s).

All subject matter of the Related Applications and of any and allparent, grandparent, great-grandparent, etc. applications of the RelatedApplications is incorporated herein by reference to the extent suchsubject matter is not inconsistent herewith.

TECHNICAL FIELD

The application discloses apparatus, methods, and systems that mayrelate to electromagnetic responses that include negatively-refractivefocusing and sensing of electromagnetic energy.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 depicts a first configuration of a negatively-refractive focusingstructure.

FIG. 2 depicts a first coordinate transformation.

FIG. 3 depicts a second configuration of a negatively-refractivefocusing structure.

FIG. 4 depicts a second coordinate transformation.

FIG. 5 depicts a negatively-refractive focusing structure with an inputsurface region.

FIG. 6 depicts a first process flow.

FIG. 7 depicts a second process flow.

FIG. 8 depicts a system that includes a focusing unit and a controller.

DETAILED DESCRIPTION

In the following detailed description, reference is made to theaccompanying drawings, which form a part hereof. In the drawings,similar symbols typically identify similar components, unless contextdictates otherwise. The illustrative embodiments described in thedetailed description, drawings, and claims are not meant to belirniting. Other embodiments may be utilized, and other changes may bemade, without departing from the spirit or scope of the subject matterpresented here.

Transformation optics is an emerging field of electromagneticengineering. Transformation optics devices include lenses that refractelectromagnetic waves, where the refraction imitates the bending oflight in a curved coordinate space (a “transformation” of a flatcoordinate space), e.g. as described in A. J. Ward and J. B. Pendry,“Refraction and geometry in Maxwell's equations,” J. Mod. Optics. 43,773 (1996), J. B. Pendry and S. A. Ramakrishna, “Focusing light usingnegative refraction,” J. Phys. [Cond. Matt.] 15, 6345 (2003), D. Schuriget al, “Calculation of material properties and ray tracing intransformation media,” Optics Express 14, 9794 (2006) (“D. Schurig et al(1)”), and in U. Leonhardt and T. G. Philbin, “General relativity inelectrical engineering,” New J. Phys. 8, 247 (2006), each of which isherein incorporated by reference. The use of the term “optics” does notimply any limitation with regards to wavelength; a transformation opticsdevice may be operable in wavelength bands that range from radiowavelengths to visible wavelengths.

A first exemplary transformation optics device is the electromagneticcloak that was described, simulated, and implemented, respectively, inJ. B. Pendry et al, “Controlling electromagnetic waves,” Science 312,1780 (2006); S. A. Cummer et al, “Full-wave simulations ofelectromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006);and D. Schurig et al, “Metamaterial electromagnetic cloak at microwavefrequencies,” Science 314, 977 (2006) (“D. Schurig et al (2)”); each ofwhich is herein incorporated by reference. See also J. Pendry et al,“Electromagnetic cloaking method,” U.S. patent application Ser. No.11/459,728, herein incorporated by reference. For the electromagneticcloak, the curved coordinate space is a transformation of a flat spacethat has been punctured and stretched to create a hole (the cloakedregion), and this transformation corresponds to a set of constitutiveparameters (electric permittivity and magnetic permeability) for atransformation medium wherein electromagnetic waves are refracted aroundthe hole in imitation of the curved coordinate space.

A second exemplary transformation optics device is illustrated byembodiments of the electromagnetic compression structure described in J.B. Pendry, D. Schurig, and D. R. Smith, “Electromagnetic compressionapparatus, methods, and systems,” U.S. patent application Ser. No.11/982,353; and in J. B. Pendry, D. Schurig, and D. R. Smith,“Electromagnetic compression apparatus, methods, and systems,” U.S.patent application Ser. No. 12/069,170; each of which is hereinincorporated by reference. In embodiments described therein, anelectromagnetic compression structure includes a transformation mediumwith constitutive parameters corresponding to a coordinatetransformation that compresses a region of space intermediate first andsecond spatial locations, the effective spatial compression beingapplied along an axis joining the first and second spatial locations.The electromagnetic compression structure thereby provides an effectiveelectromagnetic distance between the first and second spatial locationsgreater than a physical distance between the first and second spatiallocations.

A third exemplary transform optics device is illustrated by embodimentsof the electromagnetic cloaking and/or translation structure describedin J. T. Kare, “Electromagnetic cloaking apparatus, methods, andsystems,” U.S. patent application Ser. No. 12/074,247; and in J. T.Kare, “Electromagnetic cloaking apparatus, methods, and systems,” U.S.patent application Ser. No. 12/074,248; each of which is hereinincorporated by reference. In embodiments described therein, anelectromagnetic translation structure includes a transformation mediumthat provides an apparent location of an electromagnetic transducerdifferent then an actual location of the electromagnetic transducer,where the transformation medium has constitutive parameterscorresponding to a coordinate transformation that maps the actuallocation to the apparent location. Alternatively or additionally,embodiments include an electromagnetic cloaking structure operable todivert electromagnetic radiation around an obstruction in a field ofregard of the transducer (and the obstruction can be anothertransducer).

Additional exemplary transformation optics devices are described in D.Schurig et al, “Transformation-designed optical elements,” Opt. Exp. 15,14772 (2007); M. Rahm et al, “Optical design of reflectionless complexmedia by finite embedded coordinate transformations,” Phys. Rev. Lett.100, 063903 (2008); and A. Kildishev and V. Shalaev, “Engineering spacefor light via transformation optics,” Opt. Lett. 33, 43 (2008); each ofwhich is herein incorporated by reference.

In general, for a selected coordinate transformation, a transformationmedium can be identified wherein electromagnetic waves refract as ifpropagating in a curved coordinate space corresponding to the selectedcoordinate transformation. Constitutive parameters. of thetransformation medium can be obtained from the equations:

{tilde over (ε)} ^(i′j′) =|det (Λ)|⁻¹Λ_(i) ^(i′)Λ_(j) ^(j′)ε^(ij)   (1)

{tilde over (μ)}^(i′j′) =|det(Λ)|⁻¹Λ_(i) ^(i′)Λ_(j) ^(j′)μ^(ij)   (2)

where {tilde over (ε)} and {tilde over (μ)} are the permittivity andpermeability tensors of the transformation medium, ε and μ are thepermittivity and permeability tensors of the original medium in theuntransformed coordinate space, and

$\begin{matrix}{\Lambda_{i}^{i^{\prime}} = \frac{\partial x^{i^{\prime}}}{\partial x^{i}}} & (3)\end{matrix}$

is the Jacobian matrix corresponding to the coordinate transformation.In some applications, the coordinate transformation is a one-to-onemapping of locations in the untransformed coordinate space to locationsin the transformed coordinate space, and in other applications thecoordinate transformation is a one-to-many mapping of locations in theuntransformed coordinate space to locations in the transformedcoordinate space. Some coordinate transformations, such as one-to-manymappings, may correspond to a transformation medium having a negativeindex of refraction. In some applications, only selected matrix elementsof the permittivity and permeability tensors need satisfy equations (1)and (2), e.g. where the transformation optics response is for a selectedpolarization only. In other applications, a first set of permittivityand permeability matrix elements satisfy equations (1) and (2) with afirst Jacobian A, corresponding to a first transformation opticsresponse for a first polarization of electromagnetic waves, and a secondset of permittivity and permeability matrix elements, orthogonal (orotherwise complementary) to the first set of matrix elements, satisfyequations (1) and (2) with a second Jacobian Λ′, corresponding to asecond transformation optics response for a second polarization ofelectromagnetic waves. In yet other applications, reduced parameters areused that may not satisfy equations (1) and (2), but preserve productsof selected elements in (1) and selected elements in (2), thuspreserving dispersion relations inside the transformation medium (see,for example, D. Schurig et al (2), supra, and W. Cai et al, “Opticalcloaking with metamaterials,” Nature Photonics 1, 224 (2007), hereinincorporated by reference). Reduced parameters can be used, for example,to substitute a magnetic response for an electric response, or viceversa. While reduced parameters preserve dispersion relations inside thetransformation medium (so that the ray or wave trajectories inside thetransformation medium are unchanged from those of equations (1) and(2)), they may not preserve impedance characteristics of thetransformation medium, so that rays or waves incident upon a boundary orinterface of the transformation medium may sustain reflections (whereasin general a transformation medium according to equations (1) and (2) issubstantially nonreflective). The reflective or scatteringcharacteristics of a transformation medium with reduced parameters canbe substantially reduced or eliminated by a suitable choice ofcoordinate transformation, e.g. by selecting a coordinate transformationfor which the corresponding Jacobian A (or a subset of elements thereof)is continuous or substantially continuous at a boundary or interface ofthe transformation medium (see, for example, W. Cai et al, “Nonmagneticcloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (2007),herein incorporated by reference).

In general, constitutive parameters (such as permittivity andpermeability) of a medium responsive to an electromagnetic wave can varywith respect to a frequency of the electromagnetic wave (orequivalently, with respect to a wavelength of the electromagnetic wavein vacuum or in a reference medium). Thus, a medium can haveconstitutive parameters ε₁, μ₁, etc. at a first frequency, andconstitutive parameters ε₂, μ₂, etc. at a second frequency; and so onfor a plurality of constitutive parameters at a plurality offrequencies. In the context of a transformation medium, constitutiveparameters at a first frequency can provide a first response toelectromagnetic waves at the first frequency, corresponding to a firstselected coordinate transformation, and constitutive parameters at asecond frequency can provide a second response to electromagnetic wavesat the second frequency, corresponding to a second selected coordinatetransformation; and so on: a plurality of constitutive parameters at aplurality of frequencies can provide a plurality of responses toelectromagnetic waves corresponding to a plurality of coordinatetransformations. In some embodiments the first response at the firstfrequency is substantially nonzero (i.e. one or both of ε₁ and μ₁ issubstantially non-unity), corresponding to a nontrivial coordinatetransformation, and a second response at a second frequency issubstantially zero (i.e. ε₂ and μ₂ are substantially unity),corresponding to a trivial coordinate transformation (i.e. a coordinatetransformation that leaves the coordinates unchanged); thus,electromagnetic waves at the first frequency are refracted(substantially according to the nontrivial coordinate transformation),and electromagnetic waves at the second frequency are substantiallynonrefracted. Constitutive parameters of a medium can also change withtime (e.g. in response to an external input or control signal), so thatthe response to an electromagnetic wave can vary with respect tofrequency and/or time. Some embodiments may exploit this variation withfrequency and/or time to provide respective frequency and/or timemultiplexing/demultiplexing of electromagnetic waves. Thus, for example,a transformation medium can have a first response at a frequency at timet₁, corresponding to a first selected coordinate transformation, and asecond response at the same frequency at time t₂, corresponding to asecond selected coordinate transformation. As another example, atransformation medium can have a response at a first frequency at timet₁, corresponding to a selected coordinate transformation, andsubstantially the same response at a second frequency at time t₂. In yetanother example, a transformation medium can have, at time t₁, a firstresponse at a first frequency and a second response at a secondfrequency, whereas at time t₂, the responses are switched, i.e., thesecond response (or a substantial equivalent thereof) is at the firstfrequency and the first response (or a substantial equivalent thereof)is at the second frequency. The second response can be a zero orsubstantially zero response. Other embodiments that utilize frequencyand/or time dependence of the transformation medium will be apparent toone of skill in the art.

Constitutive parameters such as those of equations (1) and (2) (orreduced parameters derived therefrom) can be realized usingartificially-structured materials. Generally speaking, theelectromagnetic properties of artificially-structured materials derivefrom their structural configurations, rather than or in addition totheir material composition.

In some embodiments, the artificially-structured materials are photoniccrystals. Some exemplary photonic crystals are described in J. D.Joannopoulos et al, Photonic Crystals: Molding the Flow of Light, 2^(nd)Edition, Princeton Univ. Press, 2008, herein incorporated by reference.In photonic crystals, photonic dispersion relations and/or photonic bandgaps are engineered by imposing a spatially-varying pattern on anelectromagnetic material (e.g. a conducting, magnetic, or dielectricmaterial) or a combination of electromagnetic materials. The photonicdispersion relations translate to effective constitutive parameters(e.g. permittivity, permeability, index of refraction) for the photoniccrystal. The spatially-varying pattern is typically periodic,quasi-periodic, or colloidal periodic, with a length scale comparable toan operating wavelength of the photonic crystal.

In other embodiments, the artificially-structured materials aremetamaterials. Some exemplary metamaterials are described in R. A. Hydeet al, “Variable metamaterial apparatus,” U.S. patent application Ser.No. 11/355,493; D. Smith et al, “Metamaterials,” InternationalApplication No. PCT/US2005/026052; D. Smith et al, “Metamaterials andnegative refractive index,” Science 305, 788 (2004); D. Smith et al,“Indefinite materials,” U.S. patent application Ser. No. 10/525,191; C.Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission LineTheory and Microwave Applications, Wiley-Interscience, 2006; N. Enghetaand R. W. Ziolkowski, eds., Metamaterials: Physics and EngineeringExplorations, Wiley-Interscience, 2006; and A. K. Sarychev and V. M.Shalaev, Electrodynamics of Metamaterials, World Scientific, 2007; eachof which is herein incorporated by reference.

Metamaterials generally feature subwavelength elements, i.e. structuralelements with portions having electromagnetic length scales smaller thanan operating wavelength of the metamaterial, and the subwavelengthelements have a collective response to electromagnetic radiation thatcorresponds to an effective continuous medium response, characterized byan effective permittivity, an effective permeability, an effectivemagnetoelectric coefficient, or any combination thereof. For example,the electromagnetic radiation may induce charges and/or currents in thesubwavelength elements, whereby the subwavelength elements acquirenonzero electric and/or magnetic dipole moments. Where the electriccomponent of the electromagnetic radiation induces electric dipolemoments, the metamaterial has an effective permittivity; where themagnetic component of the electromagnetic radiation induces magneticdipole moments, the metamaterial has an effective permeability; andwhere the electric (magnetic) component induces magnetic (electric)dipole moments (as in a chiral metamaterial), the metamaterial has aneffective magnetoelectric coefficient. Some metamaterials provide anartificial magnetic response; for example, split-ring resonators(SRRs)—or other LC or plasmonic resonators—built from nonmagneticconductors can exhibit an effective magnetic permeability (c.f. J. B.Pendry et al, “Magnetism from conductors and enhanced nonlinearphenomena,” IEEE Trans. Micro. Theo. Tech. 47, 2075 (1999), hereinincorporated by reference). Some metamaterials have “hybrid”electromagnetic properties that emerge partially from structuralcharacteristics of the metamaterial, and partially from intrinsicproperties of the constituent materials. For example, G. Dewar, “A thinwire array and magnetic host structure with n<0,” J. Appl. Phys. 97,10Q101 (2005), herein incorporated by reference, describes ametamaterial consisting of a wire array (exhibiting a negativepermeability as a consequence of its structure) embedded in anonconducting ferrimagnetic host medium (exhibiting an intrinsicnegative permeability). Metamaterials can be designed and fabricated toexhibit selected permittivities, permeabilities, and/or magnetoelectriccoefficients that depend upon material properties of the constituentmaterials as well as shapes, chiralities, configurations, positions,orientations, and couplings between the subwavelength elements. Theselected permittivites, permeabilities, and/or magnetoelectriccoefficients can be positive or negative, complex (having loss or gain),anisotropic, variable in space (as in a gradient index lens), variablein time (e.g. in response to an external or feedback signal), variablein frequency (e.g. in the vicinity of a resonant frequency of themetamaterial), or any combination thereof. The selected electromagneticproperties can be provided at wavelengths that range from radiowavelengths to infrared/visible wavelengths; the latter wavelengths areattainable, e.g., with nanostructured materials such as nanorod pairs ornano-fishnet structures (c.f. S. Linden et al, “Photonic metamaterials:Magnetism at optical frequencies,” IEEE J. Select. Top. Quant. Elect.12, 1097 (2006) and V. Shalaev, “Optical negative-index metamaterials,”Nature Photonics 1, 41 (2007), both herein incorporated by reference).An example of a three-dimensional metamaterial at optical frequencies,an elongated-split-ring “woodpile” structure, is described in M. S. Rillet al, “Photonic metamaterials by direct laser writing and silverchemical vapour deposition,” Nature Materials advance onlinepublication, May 11, 2008, (doi:10.1038/nmat2197).

While many exemplary metamaterials are described as including discreteelements, some implementations of metamaterials may include non-discreteelements or structures. For example, a metamaterial may include elementscomprised of sub-elements, where the sub-elements are discretestructures (such as split-ring resonators, etc.), or the metamaterialmay include elements that are inclusions, exclusions, layers, or othervariations along some continuous structure (e.g. etchings on asubstrate). Some examples of layered metamaterials include: a structureconsisting of alternating doped/intrinsic semiconductor layers (cf. A.J. Hoffman, “Negative refraction in semiconductor metamaterials,” NatureMaterials 6, 946 (2007), herein incorporated by reference), and astructure consisting of alternating metal/dielectric layers (cf. A.Salandrino and N. Engheta, “Far-field subdiffraction optical microscopyusing metamaterial crystals: Theory and simulations,” Phys. Rev. B 74,075103 (2006); and Z. Jacob et al, “Optical hyperlens: Far-field imagingbeyond the diffraction limit,” Opt. Exp. 14, 8247 (2006); each of whichis herein incorporated by reference). The metamaterial may includeextended structures having distributed electromagnetic responses (suchas distributed inductive responses, distributed capacitive responses,and distributed inductive-capacitive responses). Examples includestructures consisting of loaded and/or interconnected transmission lines(such as microstrips and striplines), artificial ground plane structures(such as artificial perfect magnetic conductor (PMC) surfaces andelectromagnetic band gap (EGB) surfaces), and interconnected/extendednanostructures (nano-fishnets, elongated SRR woodpiles, etc.).

With reference now to FIG. 1, an illustrative embodiment is depictedthat includes a negatively-refractive focusing structure 110. This andother drawings, unless context dictates otherwise, can represent aplanar view of a three-dimensional embodiment, or a two-dimensionalembodiment (e.g. in FIG. 1 where the structures are positioned inside ametallic or dielectric slab waveguide oriented normal to the page). Thenegatively-refractive focusing structure receives and negativelyrefracts electromagnetic energy, depicted as solid rays 102 (the use ofa ray description, in FIG. 1 and elsewhere, is a heuristic conveniencefor purposes of visual illustration, and is not intended to connote anylimitations or assumptions of geometrical optics; further, the elementsdepicted in FIG. 1 can have spatial dimensions that are variously lessthan, greater than, or comparable to a wavelength of interest). Thenegatively-refracted electromagnetic energy converges towards aninterior focusing region 120 positioned inside the negatively-refractivefocusing structure 110; in this example, the interior focusing region120 is depicted as a slab having a thickness equal an axial extent 122of the interior focusing region. The axial extent 122 corresponds to anaxial direction indicated by the axial unit vector 160, with transverseunit vectors 161 and 162 defined perpendicular thereto. In FIG. 1, theelectromagnetic energy radiates from exemplary electromagnetic sources101 in an exterior field region 130, the exterior field region beingpositioned outside the negatively-refractive focusing structure. In thisexample, the exterior field region 130 is depicted as a slab having athickness equal to an axial extent 132 of the exterior field region. Theelectromagnetic sources 101 in the exterior field region correspond toelectromagnetic images 103 in the interior focusing region. The axialextent 122 of the interior focusing region (spanned, in this example, bythe electromagnetic images 103) exceeds the axial extent 132 of theexterior field region (spanned, in this example, by the electromagneticsources 101), thus demonstrating that the negatively-refractive focusingstructure provides an axial magnification greater than one, and in thisexample the axial magnification corresponds to a ratio of the axialextent of the interior focusing region to the axial extent of theexterior field region. Embodiments optionally include one or moreelectromagnetic sensors (schematically depicted as ellipses 150)positioned within the interior focusing region (in the presentedillustrative representation, sensors are linearly positioned along theaxial extent 122 of the interior focusing region, but this is notintended to be limiting).

In general, embodiments provide a negatively-refractive focusingstructure having an interior focusing region and an exterior fieldregion; electromagnetic energy that radiates from the exterior fieldregion, and couples to the negatively-refractive focusing structure, issubsequently substantially concentrated in the interior focusing region.For example, in some applications each object point in the exteriorfield region defines a point spread function and a correspondingenclosed energy region (e.g. a region wherein some selectedfraction—such as 50%, 75%, or 90%/—of electromagnetic energy thatradiates from the object point is concentrated), and the interiorfocusing region is a union of the enclosed energy regions for the objectpoints that compose the exterior field region. The negatively-refractivefocusing structure provides an axial magnification, and in someapplications the axial magnification corresponds to a ratio where thedivisor is an axial separation between first and second object pointsand the dividend is an axial separation between centroids of first andsecond point spread functions corresponding to the first and secondobject points. In some embodiments, the interior focusing region may bea planar or substantially planar slab (e.g. 120 in FIG. 1) having a slabthickness providing the axial extent of the interior focusing region(e.g. 122 in FIG. 1). In other embodiments, the interior focusing regionmay be a non-planar slab-like region, e.g. a cylindrically-,spherically-, ellipsoidally-, or otherwise-curved slab having a slabthickness providing the axial extent of the interior focusing region. Inother embodiments, the interior focusing region may be neither planarnor slab-like. In some embodiments the negatively-refractive focusingstructure defines an optical axis as a symmetry or central axis of thenegatively-refractive focusing structure, and the optical axis providesan axial direction, with transverse directions defined perpendicularthereto. More generally, one may define an axial direction correspondingto an axial extent of the interior focusing region, with transversedirections defined perpendicular thereto. This is consistent with FIG.1, where the interior focusing region is a planar slab, and the axialdirection corresponds to a unit vector normal to the slab. Where theinterior focusing region is curved, the axial direction can vary alongthe transverse extent of the focusing region. For example, where theinterior focusing region is a cylindrically- or spherically-curved slab,the axial direction corresponds to a radius unit vector (and thetransverse directions correspond to height/azimuth unit vectors orazimuth/zenith unit vectors, respectively); where the interior focusingregion is an otherwise-curved slab, the axial direction corresponds to avector locally normal to the slab surface (and the transverse directionscorrespond to orthogonal unit vectors locally tangent to the slabsurface).

In some embodiments a negatively-refractive focusing structure, such asthat depicted in FIG. 1, includes a transformation medium. For example,the ray trajectories 102 in FIG. 1 correspond to a coordinatetransformation that is multiple-valued and includes both a coordinateinversion and a uniform spatial dilation along the axial direction 160(within the axial extent of the negatively-refractive focusing structure110); this coordinate transformation can be used to identifyconstitutive parameters for a corresponding transformation medium (e.g.as provided in equations (1) and (2), or reduced parameters obtainedtherefrom) that responds to electromagnetic radiation as in FIG. 1.Explicitly, for the example of FIG. 1, defining z as an untransformedaxial coordinate and z′ as a transformed axial coordinate (where theaxial coordinates are measured along the axial direction 160), themultiple-valued coordinate transformation is depicted in FIG. 2, withfirst, second, and third branches 201, 202, and 203 corresponding tofunctions z′=f₁(z), z′=f₂(z), and z′=f₃(z), respectively. The firstbranch 201 is an identity transformation (f₁(z)=z) and maps anuntransformed coordinate region 220 to the exterior field region 130.The second branch 202 includes an axial coordinate inversion and auniform axial coordinate dilation, and maps the untransformed coordinateregion 220 to the interior focusing region 120. The third branch 203 isa shifted itentity transformation (f₃(z)=z+C, where C is a constant).The figure also indicates the axial extent of the negatively-refractivefocusing structure 110 on the z′-axis (coinciding, in this example, withthe range of the second branch 202). On the second branch, defining ascale factor

$\begin{matrix}{{s = {\frac{z^{\prime}}{z} = {f_{2}^{\prime}(z)}}},} & (4)\end{matrix}$

the example of FIGS. 1-2 presents a constant negative scale factor s<−1within the negatively-refractive focusing structure 110, correspondingto a coordinate inversion (whereby s<0) and a uniform spatial dilation(whereby |s|>1; in some instances within this document, as shall beapparent to one of skill in the art, the use of the term “scale factor,”when used in the context of a spatial dilation, may refer to theabsolute value of a negative scale factor such as described here).Supposing that the negatively-refractive focusing structure issurrounded by an ambient isotropic medium with constitutive parametersε^(ij)=εδ^(ij), μ_(ij)=μδ^(ij) (where δ^(ij) denotes the Kroneckerdelta-function, with δ^(ij)=1 for i=j and δ^(ij)=0 for i≠j), theconstitutive parameters of the transformation medium are obtained fromequations (1) and (2) and are given by (in a basis with unit vectors161, 162, and 160, respectively, in FIG. 1)

$\begin{matrix}{{\overset{\sim}{ɛ} = {\begin{pmatrix}s^{- 1} & 0 & 0 \\0 & s^{- 1} & 0 \\0 & 0 & s\end{pmatrix}ɛ}},\mspace{14mu} {\overset{\sim}{\mu} = {\begin{pmatrix}s^{- 1} & 0 & 0 \\0 & s^{- 1} & 0 \\0 & 0 & s\end{pmatrix}{\mu.}}}} & (5)\end{matrix}$

Thus, the uniform spatial dilation of FIGS. 1-2 corresponds to atransformation medium that is a uniform uniaxial medium. Moreover, thescale factor is negative, so that the constitutive parameters inequation (5) are negative, and the transformation medium is anegatively-refractive medium defining a negative index of refraction.

In some embodiments, the negatively-refractive focusing structureincludes a transformation medium that provides a non-uniform spatialdilation. An example is depicted in FIG. 3 and the correspondingmultiple-valued coordinate transformation is depicted in FIG. 4. In FIG.3, as in FIG. 1, a negatively-refractive focusing structure 110 providesan interior focusing region 120 for electromagnetic energy that radiatesfrom an exterior field region 130. In contrast to FIG. 1, however, theembodiment of FIG. 3 provides a non-uniform scale factor s (the slope ofthe mapping function z′=f₂ (z) for the second branch 202 of themultiple-valued coordinate transformation); indeed, the scale factor inthis relation satisfies, in some interval(s), the relation −1<s<0(corresponding to a local spatial compression and coordinate inversion),and in other interval(s), the relation s<−1 (corresponding to a localspatial dilation and coordinate inversion). The constitutive relationsare again given by equations (5), where s is variable in the axialdirection, and the transformation medium is a non-uniform uniaxialmedium (again with negative constitutive parameters and defining anegative index of refraction).

More generally, embodiments of a negatively-refractive focusingstructure, operable to provide an interior focusing region forelectromagnetic energy that radiates from an exterior field region, maycomprise a transformation medium, the transformation mediumcorresponding to a multiple-valued coordinate transformation that mapsan untransformed region to the exterior field region, and further mapsthe untransformed region to the interior focusing region; and theconstitutive relations of this transformation medium may be implementedwith an artificially-structured material (such as a metamaterial), asdescribed previously. In some embodiments, the coordinate transformationincludes a coordinate inversion and spatial dilation along an axialdirection of the interior focusing region, and a scale factor of thespatial dilation (within the interior focusing region) may correspond toa ratio of an axial extent of the interior focusing region to an axialextent of the exterior field region. This is consistent with FIGS. 2 and4, where the slope triangle 200, indicating a scale factor in theinterior focusing region, is similar or substantially similar to atriangle with a base 220 (equal to 130, for a first branch 201 that isan identity transformation) and a height 120. Just as the axialdirection can vary along a transverse extent of the interior focusingregion, the direction of the coordinate inversion/dilation can vary aswell. Thus, for example, a substantially cylindrically- orspherically-curved interior focusing region may correspond to a (uniformor non-uniform) inversion/dilation of a cylindrical or spherical radiuscoordinate; a substantially ellipsoidally-curved interior focusingregion may correspond to a (uniform or non-uniform) inversion/dilationof a confocal ellipsoidal coordinate; etc.

The negatively-refractive focusing structure 110 is depicted in FIGS. 1and 3 as a planar slab, but this is a schematic illustration and is notintended to be limiting. In various embodiments thenegatively-refractive focusing structure can be a cylindrically-,spherically-, or ellipsoidally-curved slab, or any other slab- ornon-slab-like structure configured to provide an interior focusingregion for negatively-refracted electromagnetic energy with an axialmagnification substantially greater than one. Some embodiments, such asthat depicted in FIG. 5, define an input surface region 510 as a surfaceregion of the negatively-refractive focusing structure 110 that receiveselectromagnetic radiation from an adjacent region 500, and this inputsurface region may be substantially nonreflective of the receivedelectromagnetic radiation. For example, where the negatively-refractivefocusing structure is a transformation medium, equations (1) and (2)generally provide a medium that is substantially nonreflective. Moregenerally, the input surface region may be substantially nonreflectiveby virtue of a substantial impedance-matching to the adjacent region.With impedance-matching, a wave impedance of the input surface region issubstantially equal to a wave impedance of the adjacent region. The waveimpedance of an isotropic medium is

$\begin{matrix}{Z_{0} = \sqrt{\frac{\mu}{ɛ}}} & (6)\end{matrix}$

while the wave impedance of a generally anisotropic medium is a tensorquantity, e.g. as defined in L. M. Barkovskii and G. N. Borzdov, “Theimpedance tensor for electromagnetic waves in anisotropic media,” J.Appl. Spect. 20, 836 (1974) (herein incorporated by reference). In someembodiments an impedance-matching is a substantial matching of everymatrix element of the wave impedance tensor (i.e. to provide asubstantially nonreflective interface for all incident polarizations);in other embodiments an impedance-matching is a substantial matching ofonly selected matrix elements of the wave impedance tensor (e.g. toprovide a substantially nonreflective interface for a selectedpolarization only). In some embodiments, the adjacent region defines apermittivity ε₁ and a permeability μ₁, where either or both parametersmay be substantially unity or substantially non-unity; the input surfaceregion defines a permittivity ε₂ and a permeability μ₂, where either orboth parameters may be substantially unity or substantially non-unity;and the impedance-matching condition implies

$\begin{matrix}{\frac{ɛ_{2}}{ɛ_{1}} \cong \frac{\mu_{2}}{\mu_{1}}} & (7)\end{matrix}$

where ε₂ and μ₂ may be tensor quantities. Defining a surface normaldirection and a surface parallel direction (e.g. depicted as elements521 and 522, respectively, in FIG. 5), some embodiments provide a inputsurface region that defines: a surface normal permittivity ε 1/2corresponding to the surface normal direction and a surface parallelpermittivity ε 1/2 corresponding to the surface parallel direction;and/or a surface normal permeability μ 1/2 corresponding to the surfacenormal direction and a surface parallel permeability μ 1/2 correspondingto the surface parallel direction; and the impedance-matching conditionmay imply (in addition to equation (7)) one or both of the followingconditions:

$\begin{matrix}{{\frac{ɛ_{2}^{\bot}}{ɛ_{1}} \cong \frac{ɛ_{1}}{ɛ_{2}^{\parallel}}}\;,\mspace{11mu} {\frac{\mu_{2}^{\bot}}{\mu_{1}} \cong {\frac{\mu_{1}}{\mu_{2}^{\parallel}}.}}} & (8)\end{matrix}$

Where the input surface region is a curved surface region (e.g. as inFIG. 5), the surface normal direction and the surface parallel directioncan vary with position along the input surface region.

Some embodiments provide one or more electromagnetic sensors positionedwithin the interior focusing region of the negatively-refractivefocusing structure. In general, electromagnetic sensors, such as thosedepicted FIG. 1 and in other embodiments, are electromagnetic deviceshaving a detectable response to received or absorbed electromagneticenergy. Electromagnetic sensors can include antennas (such as wire/loopantennas, horn antennas, reflector antennas, patch antennas, phasedarray antennas, etc.), solid-state photodetectors (such as photodiodes,CCDs, and photoresistors), vacuum photodetectors (such as phototubes andphotomultipliers) chemical photodetectors (such as photographicemulsions), cryogenic photodetectors (such as bolometers),photoluminescent detectors (such as phosphor powders or fluorescentdyes/markers), MEMS detectors (such as microcantilever arrays withelectromagnetically responsive materials or elements) or any otherdevices operable to detect and/or transduce electromagnetic energy. Someembodiments include a plurality of electromagnetic sensors positionedwithin the interior focusing region. A first example is a multiplet ofsensors operable at a corresponding multiplet of wavelengths orwavelength bands, i.e. a first sensor operable at a firstwavelength/wavelength band, a second sensor operable at a secondwavelength/wavelength band, etc. A second example is a focal plane arrayof sensors or sensor multiplets (e.g. a Bayer or Foveon sensor). A thirdexample is a phased array of antennas. The plurality of sensors can beaxially distributed (as in FIG. 1); for example, the axial extent of theinterior focusing region may admit a plurality of parallel focal planesensor arrays.

In some embodiments the negatively-refractive focusing structureprovides an interior focusing region with an axial magnificationsubstantially greater than one for electromagnetic energy at a selectedfrequency/frequency band and/or a selected polarization. The selectedfrequency or frequency band may be selected from a range that includesradio frequencies, microwave frequencies, millimeter- orsubmillimeter-wave frequencies, THz-wave frequencies, opticalfrequencies (e.g. variously corresponding to soft x-rays, extremeultraviolet, ultraviolet, visible, near-infrared, infrared, or farinfrared light), etc. The selected polarization may be a particular TEpolarization (e.g. where the electric field is in a particular directiontransverse to the axial direction, as with s-polarized electromagneticenergy), a particular TM polarization (e.g. where the magnetic field isin a particular direction transverse to the axial direction, as withp-polarized electromagnetic energy), a circular polarization, etc.(other embodiments provide an interior focusing region with an axialmagnification substantially greater than one that is substantially thesame interior focusing region with substantially the same axialmagnification for any polarization of electromagnetic energy, e.g. forunpolarized electromagnetic energy).

In other embodiments the negatively-refractive focusing structureprovides a first interior focusing region with a first axialmagnification substantially greater than one for electromagnetic energyat a first frequency, and a second interior focusing region with asecond axial magnification substantially greater than one forelectromagnetic energy at a second frequency. The first axialmagnification may be different than or substantially equal to the firstaxial magnification, and the first and second interior focusing regionsmay be substantially (or completely) non-overlapping, partiallyoverlapping or substantially (or completely) overlapping. Forembodiments that recite first and second frequencies, the first andsecond frequencies may be selected from the frequency categories in thepreceding paragraph. Moreover, for these embodiments, the recitation offirst and second frequencies may generally be replaced by a recitationof first and second frequency bands, again selected from the abovefrequency categories. These embodiments providing anegatively-refractive focusing structure operable at first and secondfrequencies may include a transformation medium having an adjustableresponse to electromagnetic radiation. For example, the transformationmedium may have a response to electromagnetic radiation that isadjustable (e.g. in response to an external input or control signal)between a first response and a second response, the first responseproviding the first interior focusing region for electromagnetic energyat the first frequency, and the second response providing the secondinterior focusing region for electromagnetic energy at the secondfrequency. A transformation medium with an adjustable electromagneticresponse may be implemented with variable metamaterials, e.g. asdescribed in R. A. Hyde et al, supra. Other embodiments of anegatively-refractive focusing structure operable at first and secondfrequencies may include transformation medium having afrequency-dependent response to electromagnetic radiation, correspondingto frequency-dependent constitutive parameters. For example, thefrequency-dependent response at a first frequency may provide a firstinterior focusing region for electromagnetic energy at the firstfrequency, and the frequency-dependent response at a second frequencymay provide second interior focusing region for electromagnetic energyat the second frequency. A transformation medium having afrequency-dependent response to electromagnetic radiation can beimplemented with artificially-structured materials such asmetamaterials; for example, a first set of metamaterial elements havinga response at the first frequency may be interleaved with a second setof metamaterial elements having a response at the second frequency.

An illustrative embodiment is depicted as a process flow diagram in FIG.6.

Flow 600 includes operation 610—negatively refracting an electromagneticwave at a surface region, the surface region defining a surface normaldirection. For example, a negatively-refractive focusing structure, suchas that depicted as element 110 in FIG. 5, may include an input surfaceregion 510 that negatively refracts electromagnetic energy incident uponthe input surface region from an adjacent region, and thenegatively-refractive focusing structure may include a transformationmedium that provides a coordinate inversion (for a coordinatecorresponding to a direction normal to the surface, e.g. the direction521 in FIG. 5), the coordinate inversion corresponding to anegatively-refractive response of the transformation medium. Flow 600further includes operation 620—spatially dilating the refractedelectromagnetic wave along a dilation direction, the dilation directioncorresponding to the surface normal direction. For example, anegatively-refractive focusing structure, such as that depicted aselement 110 in FIGS. 1 and 3, may spatially dilate refractedelectromagnetic energy 102 along an axial direction (e.g. the direction160 in FIGS. 1 and 3) to provide an axial extent of an interior focusingregion 120 greater than an axial extent of an exterior field region 130(the ratio of axial extents corresponding to a provided axialmagnification), and the negatively-refractive focusing structure mayinclude a transformation medium that provides a coordinate dilation (foran axial coordinate corresponding to the axial direction 160), thecoordinate dilation having a scale factor corresponding to the providedaxial magnification. Operation 620 optionally includes sub-operation622—spatially dilating a first component of the refractedelectromagnetic wave at a first frequency along the dilationdirection-and sub-operation 624—spatially dilating a second component ofthe refracted electromagnetic wave at a second frequency along thedilation direction. For example, a negatively-refractive focusingstructure may provide a first interior focusing region with a firstaxial magnification for electromagnetic energy at a first frequency anda second interior focusing region with a second axial magnification forelectromagnetic energy at a second frequency, where the second axialmagnification may be different than or substantially equal to the firstaxial magnification; and this negatively-refractive focusing structureoperable at first and second frequencies may include a transformationmedium having an adjustable response to electromagnetic radiation, or atransformation medium having a frequency-dependent response toelectromagnetic radiation. Flow 600 optionally includes operation630—sensing the electromagnetic wave at one or more locations within afocusing region that is provided by the negatively refracting and thespatially dilating. For example, a negatively-refractive focusingstructure 110, such as that depicted in FIGS. 1 and 3, may provide aninterior focusing region 120, and one or more electromagnetic sensors,such as those depicted as elements 150 in FIG. 1, may be positionedwithin the interior focusing region to detect/receive/absorb theelectromagnetic energy 102.

Another illustrative embodiment is depicted as a process flow diagram inFIG. 7. Flow 700 includes operation 710—determining electromagneticparameters that define a negative refractive index in a spatial region,the electromagnetic parameters providing an axial magnificationsubstantially greater than one for an interior focusing region withinthe spatial region. For example, the spatial region may be a volume thatencloses a negatively-refractive focusing structure, such as thatdepicted as element 110 in FIGS. 1 and 3, and the determinedelectromagnetic parameters may be the electromagnetic parameters of thenegatively-refractive focusing structure. The negatively-refractivefocusing structure may include a transformation medium, where thedetermined electromagnetic parameters satisfy or substantially satisfyequations (1) and (2), as described above; or, the determinedelectromagnetic parameters may be reduced parameters (as discussedearlier) where the corresponding non-reduced parameters satisfyequations (1) and (2). In some embodiments, the determining of theelectromagnetic parameters includes: determining a coordinatetransformation (such as those depicted in FIGS. 2 and 4); thendetermining electromagnetic parameters for a correspondingtransformation medium (e.g. with equations (1) and (2)); then,optionally, reducing the electromagnetic parameters (e.g. to at leastpartially substitute a magnetic response for an electromagneticresponse, or vice versa, as discussed above). Operation 710 optionallyincludes sub-operation 712—for electromagnetic waves at a firstfrequency, determining a first subset of the electromagnetic parametersproviding a first axial magnification substantially greater than one fora first interior focusing subregion within the interior focusingregion-and sub-operation 714—for electromagnetic waves at a secondfrequency, determining a second subset of the electromagnetic parametersproviding a second axial magnification substantially greater than onefor a second interior focusing subregion within the interior focusingregion. For example, the determined electromagnetic parameters may bethe electromagnetic parameters of a negatively-refractive focusingstructure providing a first interior focusing region with a first axialmagnification for electromagnetic energy at a first frequency and asecond interior focusing region with a second axial magnification forelectromagnetic energy at a second frequency. The negatively-refractivefocusing structure may include a transformation medium having anadjustable response to electromagnetic radiation, e.g. adjustablebetween a first response, corresponding to the first subset of theelectromagnetic parameters, and a second response, corresponding to thesecond subset of the electromagnetic parameters. Or, thenegatively-refractive focusing structure may include a transformationmedium having a frequency-dependent response to electromagneticradiation, corresponding to frequency-dependent constitutive parameters,so that the first and second subsets of the electromagnetic parametersare values of the frequency-dependent constitutive parameters at thefirst and second frequencies, respectively. Flow 700 optionally furtherincludes operation 720—selecting one or more positions of one or moreelectromagnetic sensors within the spatial region. For example,electromagnetic sensors may be positioned in a phased array, an focalplane array, an axially-distributed arrangement, etc. Flow 700optionally further includes operation 730—configuring anartificially-structured material having an effective electromagneticresponse that corresponds to the electromagnetic parameters in thespatial region. For example, the configuring may include configuring thestructure(s) and/or the materials that compose a photonic crystal or ametamaterial. Operation 730 optionally includes determining anarrangement of a plurality of electromagnetically responsive elementshaving a plurality of individual responses, the plurality of individualresponses composing the effective electromagnetic response. For example,the determining may include determining the positions, orientations, andindividual response parameters (spatial dimensions, resonantfrequencies, linewidths, etc.) of a plurality of metamaterial elementssuch as split-ring resonators, wire or nanowire pairs, etc. Operation730 optionally includes configuring at least oneelectromagnetically-responsive structure to arrange a plurality ofdistributed electromagnetic responses, the plurality of distributedelectromagnetic responses composing the effective electromagneticresponse. For example, the configuring may include configuring thedistribution of loads and interconnections on a transmission linenetwork, configuring an arrangement of layers in a layered metamaterial,configuring a pattern of etching or deposition (as with a nano-fishnetstructure), etc.

With reference now to FIG. 8, an illustrative embodiment is depicted asa system block diagram. The system 800 includes a focusing unit 810optionally coupled to a controller unit 830. The focusing unit 810 mayinclude a negatively-refractive focusing structure such as that depictedas element 110 in FIGS. 1 and 3. The negatively-refractive focusingstructure may be a variable negatively-refractive focusing structure,such as a variable metamaterial responsive to one or more control inputsto vary one or more focusing characteristics (axial magnification,operating frequency/frequency band, operating polarization, effectivecoordinate transformation for a transformation medium, etc.); and thecontroller unit 830 may include control circuitry that provides one ormore control inputs to the variable negatively-refractive focusingstructure. The system 800 optionally further includes a sensing unit 820that may include one or more sensors, such as those depicted as elements150 in FIG. 1, and associated circuitry such as receiver circuitry,detector circuitry, and/or signal processing circuitry. The sensing unit820 is optionally coupled to the controller unit 830, and in someembodiments the controller unit 830 includes circuitry for coordinatingor synchronizing the operation of the focusing unit 810 and the sensingunit 820. The controller unit 830 may include circuitry responsive tosensor data (from the sensor unit 820) to vary the focusingcharacteristics of the focusing structure. As a first example, thecontroller unit may include circuitry responsive to the sensor data toidentify a frequency/polarization of received energy, and adjust thefocusing unit to an operating frequency/polarization substantially equalto the frequency/polarization of the received energy. As a secondexample, the controller unit may include circuitry responsive to thesensor data to identify a target interior focusing region (and/or atarget axial magnification), and corresponding adjust the focusingsystem whereby a negatively-refractive focusing structure provides aninterior focusing region substantially equal to the target interiorfocusing region (and/or an axial magnification substantially equal tothe target axial magnification).

The foregoing detailed description has set forth various embodiments ofthe devices and/or processes via the use of block diagrams, flowcharts,and/or examples. Insofar as such block diagrams, flowcharts, and/orexamples contain one or more functions and/or operations, it will beunderstood by those within the art that each function and/or operationwithin such block diagrams, flowcharts, or examples can be implemented,individually and/or collectively, by a wide range of hardware, software,firmware, or virtually any combination thereof. In one embodiment,several portions of the subject matter described herein may beimplemented via Application Specific Integrated Circuits (ASICs), FieldProgrammable Gate Arrays (FPGAs), digital signal processors (DSPs), orother integrated formats. However, those skilled in the art willrecognize that some aspects of the embodiments disclosed herein, inwhole or in part, can be equivalently implemented in integratedcircuits, as one or more computer programs running on one or morecomputers (e.g., as one or more programs running on one or more computersystems), as one or more programs running on one or more processors(e.g., as one or more programs running on one or more microprocessors),as firmware, or as virtually any combination thereof, and that designingthe circuitry and/or writing the code for the software and or firmwarewould be well within the skill of one of skill in the art in light ofthis disclosure. In addition, those skilled in the art will appreciatethat the mechanisms of the subject matter described herein are capableof being distributed as a program product in a variety of forms, andthat an illustrative embodiment of the subject matter described hereinapplies regardless of the particular type of signal bearing medium usedto actually carry out the distribution. Examples of a signal bearingmedium include, but are not limited to, the following: a recordable typemedium such as a floppy disk, a hard disk drive, a Compact Disc (CD), aDigital Video Disk (DVD), a digital tape, a computer memory, etc.; and atransmission type medium such as a digital and/or an analogcommunication medium (e.g., a fiber optic cable, a waveguide, a wiredcommunications link, a wireless communication link, etc.).

In a general sense, those skilled in the art will recognize that thevarious aspects described herein which can be implemented, individuallyand/or collectively, by a wide range of hardware, software, firmware, orany combination thereof can be viewed as being composed of various typesof “electrical circuitry.” Consequently, as used herein “electricalcircuitry” includes, but is not limited to, electrical circuitry havingat least one discrete electrical circuit, electrical circuitry having atleast one integrated circuit, electrical circuitry having at least oneapplication specific integrated circuit, electrical circuitry forming ageneral purpose computing device configured by a computer program (e.g.,a general purpose computer configured by a computer program which atleast partially carries out processes and/or devices described herein,or a microprocessor configured by a computer program which at leastpartially carries out processes and/or devices described herein),electrical circuitry forming a memory device (e.g., forms of randomaccess memory), and/or electrical circuitry forming a communicationsdevice (e.g., a modem, communications switch, or optical-electricalequipment). Those having skill in the art will recognize that thesubject matter described herein may be implemented in an analog ordigital fashion or some combination thereof.

All of the above U.S. patents, U.S. patent application publications,U.S. patent applications, foreign patents, foreign patent applicationsand non-patent publications referred to in this specification and/orlisted in any Application Data Sheet, are incorporated herein byreference, to the extent not inconsistent herewith.

One skilled in the art will recognize that the herein describedcomponents (e.g., steps), devices, and objects and the discussionaccompanying them are used as examples for the sake of conceptualclarity and that various configuration modifications are within theskill of those in the art. Consequently, as used herein, the specificexemplars set forth and the accompanying discussion are intended to berepresentative of their more general classes. In general, use of anyspecific exemplar herein is also intended to be representative of itsclass, and the non-inclusion of such specific components (e.g., steps),devices, and objects herein should not be taken as indicating thatlimitation is desired.

With respect to the use of substantially any plural and/or singularterms herein, those having skill in the art can translate from theplural to the singular and/or from the singular to the plural as isappropriate to the context and/or application. The varioussingular/plural permutations are not expressly set forth herein for sakeof clarity.

While particular aspects of the present subject matter described hereinhave been shown and described, it will be apparent to those skilled inthe art that, based upon the teachings herein, changes and modificationsmay be made without departing from the subject matter described hereinand its broader aspects and, therefore, the appended claims are toencompass within their scope all such changes and modifications as arewithin the true spirit and scope of the subject matter described herein.Furthermore, it is to be understood that the invention is defined by theappended claims. It will be understood by those within the art that, ingeneral, terms used herein, and especially in the appended claims (e.g.,bodies of the appended claims) are generally intended as “open” terms(e.g., the term “including” should be interpreted as “including but notlimited to,” the term “having” should be interpreted as “having atleast,” the term “includes” should be interpreted as “includes but isnot limited to,” etc.). It will be further understood by those withinthe art that if a specific number of an introduced claim recitation isintended, such an intent will be explicitly recited in the claim, and inthe absence of such recitation no such intent is present. For example,as an aid to understanding, the following appended claims may containusage of the introductory phrases “at least one” and “one or more” tointroduce claim recitations. However, the use of such phrases should notbe construed to imply that the introduction of a claim recitation by theindefinite articles “a” or “an” limits any particular claim containingsuch introduced claim recitation to inventions containing only one suchrecitation, even when the same claim includes the introductory phrases“one or more” or “at least one” and indefinite articles such as “a” or“an” (e.g., “a” and/or “an” should typically be interpreted to mean “atleast one” or “one or more”); the same holds true for the use ofdefinite articles used to introduce claim recitations. In addition, evenif a specific number of an introduced claim recitation is explicitlyrecited, those skilled in the art will recognize that such recitationshould typically be interpreted to mean at least the recited number(e.g., the bare recitation of “two recitations,” without othermodifiers, typically means at least two recitations, or two or morerecitations). Furthermore, in those instances where a conventionanalogous to “at least one of A, B, and C, etc.” is used, in generalsuch a construction is intended in the sense one having skill in the artwould understand the convention (e.g., “a system having at least one ofA, B, and C” would include but not be limited to systems that have Aalone, B alone, C alone, A and B together, A and C together, B and Ctogether, and/or A, B, and C together, etc.). In those instances where aconvention analogous to “at least one of A, B, or C, etc.” is used, ingeneral such a construction is intended in the sense one having skill inthe art would understand the convention (e.g., “a system having at leastone of A, B, or C” would include but not be limited to systems that haveA alone, B alone, C alone, A and B together, A and C together, B and Ctogether, and/or A, B, and C together, etc.). It will be furtherunderstood by those within the art that virtually any disjunctive wordand/or phrase presenting two or more alternative terms, whether in thedescription, claims, or drawings, should be understood to contemplatethe possibilities of including one of the terms, either of the terms, orboth terms. For example, the phrase “A or B” will be understood toinclude the possibilities of “A” or “B” or “A and B.” With respect tothe appended claims, those skilled in the art will appreciate thatrecited operations therein may generally be performed in any order.Examples of such alternate orderings may include overlapping,interleaved, interrupted, reordered, incremental, preparatory,supplemental, simultaneous, reverse, or other variant orderings, unlesscontext dictates otherwise. With respect to context, even terms like“responsive to,” “related to,” or other past-tense adjectives aregenerally not intended to exclude such variants, unless context dictatesotherwise.

While various aspects and embodiments have been disclosed herein, otheraspects and embodiments will be apparent to those skilled in the art.The various aspects and embodiments disclosed herein are for purposes ofillustration and are not intended to be limiting, with the true scopeand spirit being indicated by the following claims.

1-163. (canceled)
 164. A method, comprising: determining electromagneticparameters that define a negative refractive index in a spatial region,the electromagnetic parameters providing an axial magnificationsubstantially greater than one for an interior focusing region withinthe spatial region.
 165. (canceled)
 166. The method of claim 164,wherein the negative refractive index is greater than minus one withinat least a portion of the spatial region.
 167. The method of claim 164,wherein the negative refractive index is greater than minus one withinat least a portion of the interior focusing region 168-184. (canceled)185. The method of claim 164, further comprising: selecting one or morepositions of one or more electromagnetic sensors within the spatialregion.
 186. The method of claim 185, wherein the one or more positionsare inside the interior focusing region. 187-190. (canceled)
 191. Themethod of claim 164, wherein the electromagnetic parameters provide theinterior focusing region for electromagnetic waves that radiate from anexterior field region.
 192. The method of claim 191, wherein the axialmagnification corresponds to a ratio of an axial extent of the interiorfocusing region to an axial extent of the exterior field region. 193.The method of claim 191, wherein the electromagnetic parameters define atransformation medium.
 194. The method of claim 193, wherein thetransformation medium corresponds to a multiple-valued coordinatetransformation.
 195. The method of claim 194, wherein a first branch ofthe multiple-valued coordinate transformation maps an untransformedregion to the exterior field region, and a second branch of themultiple-valued coordinate transformation maps the untransformed regionto the interior focusing region.
 196. The method of claim 195, whereinthe first branch of the multiple-valued coordinate transformation is anidentity transformation.
 197. The method of claim 195, wherein thesecond branch of the multiple-valued coordinate transformation includesan axial coordinate inversion.
 198. The method of claim 197, wherein thesecond branch of the multiple-valued coordinate transformation includesan axial coordinate dilation.
 199. The method of claim 198, wherein theaxial magnification corresponds to a scale factor of the axialcoordinate dilation.
 200. (canceled)
 201. The method of claim 164,wherein the electromagnetic parameters include: axial electromagneticparameters; and transverse electromagnetic parameters that inverselycorrespond to the axial electromagnetic parameters.
 202. The method ofclaim 201, wherein the interior focusing region defines an axialdirection, and the axial electromagnetic parameters correspond to theaxial direction. 203-206. (canceled)
 207. The method of claim 202,wherein the transverse electromagnetic parameters correspond to atransverse direction, the transverse direction being substantiallyperpendicular to the axial direction.
 208. (canceled)
 209. (canceled)210. The method of claim 201, wherein the axial electromagneticparameters include an axial permittivity.
 211. The method of claim 210,wherein the transverse electromagnetic parameters include a transversepermittivity that is substantially a multiplicative inverse of the axialpermittivity.
 212. (canceled)
 213. The method of claim 211, wherein theaxial permittivity is less than minus one.
 214. The method of claim 210,wherein the axial electromagnetic parameters include an axialpermeability.
 215. The method of claim 214, wherein the transverseelectromagnetic parameters include a transverse permeability that issubstantially a multiplicative inverse of the axial permeability. 216.(canceled)
 217. The method of claim 215, wherein the axial permeabilityis less than minus one.
 218. The method of claim 214, wherein the axialpermittivity is substantially equal to the axial permeability.
 219. Themethod of claim 218, wherein the transverse electromagnetic parametersinclude a transverse permeability that is substantially a multiplicativeinverse of the axial permeability.
 220. (canceled)
 221. The method ofclaim 219, wherein the axial permeability is less than minus one. 222.The method of claim 164, wherein the spatial region includes an inputsurface region that is substantially nonreflective of electromagneticenergy incident upon the input surface region.
 223. The method of claim222, wherein the input surface region is substantially nonreflective ofelectromagnetic energy incident upon the input surface region from anadjacent region, the adjacent region being exterior to the spatialregion.
 224. (canceled)
 225. The method of claim 223, wherein a waveimpedance of the adjacent region is substantially equal to a waveimpedance of the input surface region.
 226. The method of claim 225,wherein the adjacent region defines a first permittivity and a firstpermeability, the input surface region defines a second permittivity anda second permeability, and a ratio of the second permittivity to thefirst permittivity is substantially equal to a ratio of the secondpermeability to the first permeability. 227-230. (canceled)
 231. Themethod of claim 226, wherein the second permittivity is less than zero.232. The method of claim 226, wherein the second permeability is lessthan zero.
 233. The method of claim 232, wherein the second permittivityis less than zero.
 234. The method of claim 226, wherein the inputsurface region defines a surface normal direction and a surface paralleldirection, the second permittivity includes a surface normalpermittivity corresponding to the surface normal direction and a surfaceparallel permittivity corresponding to the surface parallel direction,and a ratio of the surface normal permittivity to the first permittivityis substantially a multiplicative inverse of a ratio of the surfaceparallel permittivity to the first permittivity.
 235. (canceled) 236.The method of claim 234, wherein the ratio of the surface normalpermittivity to the first permittivity is less than minus one.
 237. Themethod of claim 234, wherein the second permeability includes a surfacenormal permeability corresponding to the surface normal direction and asurface parallel permeability corresponding to the surface paralleldirection, and a ratio of the surface normal permeability to the firstpermeability is substantially a multiplicative inverse of a ratio of thesurface parallel permeability to the first permeability.
 238. (canceled)239. The electromagnetic apparatus of claim 237, wherein the ratio ofthe surface normal permittivity to the first permittivity is less thanminus one.
 240. The method of claim 164, further comprising: configuringan artificially-structured material having an effective electromagneticresponse that corresponds to the electromagnetic parameters in thespatial region.
 241. The method of claim 240, wherein theartificially-structured material includes a photonic crystal.
 242. Themethod of claim 240, wherein the artificially-structured materialincludes a metamaterial.
 243. The method of claim 240; wherein theconfiguring includes: determining an arrangement of a plurality ofelectromagnetically responsive elements having a plurality of individualresponses, the plurality of individual responses composing the effectiveelectromagnetic response.
 244. The method of claim 243, wherein theplurality of individual responses includes a plurality of individualfrequency responses.
 245. The method of claim 244, wherein the pluralityof individual frequency responses is a plurality of individual frequencybandwidth responses corresponding to a plurality of individualbandwidths.
 246. The method of claim 245, wherein the plurality ofelectromagnetically responsive elements is a plurality ofelectromagnetically responsive resonators corresponding to a pluralityof resonant frequencies, and each of the individual bandwidths issubstantially centered around a corresponding one of the resonantfrequencies.
 247. (canceled)
 248. The method of claim 245, wherein theindividual bandwidths are at least partially overlapping. 249.(canceled)
 250. (canceled)
 251. The method of claim 243, wherein theelectromagnetically responsive elements include discrete circuitelements. 252-256. (canceled)
 257. The method of claim 243, wherein theelectromagnetically responsive elements include integrated circuitelements.
 258. (canceled)
 259. The method of claim 243, wherein theelectromagnetically responsive elements include metallic structures.260. The method of claim 243, wherein the electromagnetically responsiveelements include LC resonators.
 261. The method of claim 243, whereinthe electromagnetically responsive elements include plasmonicresonators.
 262. The method of claim 243, wherein theelectromagnetically responsive elements include nanostructures. 263-265.(canceled)
 266. The method of claim 243, wherein the electromagneticallyresponsive elements include split-ring resonators.
 267. (canceled) 268.The method of claim 240, wherein the configuring includes: configuringat least one electromagnetically-responsive structure to arrange aplurality of distributed electromagnetic responses, the plurality ofdistributed electromagnetic responses composing the effectiveelectromagnetic response.
 269. (canceled)
 270. (canceled)
 271. Themethod of claim 268, wherein the distributed electromagnetic responsesinclude distributed inductive-capacitive responses.
 272. The method ofclaim 268, wherein the at least one electromagnetically-responsivestructure includes transmission lines. 273-275. (canceled)